Where horses are beginning some training under direction (4)
I believe the answer is:
stud
'where horses are' is the definition.
(as in a stud farm)
'beginning some training under direction' is the wordplay.
'beginning' indicates taking the first letters.
The first letters of 'some training under direction' is 'STUD'.
(Other definitions for stud that I've seen before include "Dot" , "equine establishment" , "Wall timber" , "Part of a football boot" , "Collar button or animal farm" .)